for certain \(a \in \mathbb {R}\)  and \(b \notin \lbrace 0,-1,-2,\dots \rbrace \) . The solution \(F = F(a,b;\rho )\)  is a linear combination of the confluent hypergeometric functions of the first and second kind, \(M(a,b;\rho )\)  and \(U(a,b;\rho )\) , respectively. For more information on this ODE see Chapter 13 by Slater of the book edited by Abramowitz and Stegun [1]}.
